  3.  (9)  Suppose a pizzeria, appropriately named, “MPA = Master’s of Pizza Administration” (which has unfortunately occasionally been confused with the “Master’s of Propane Administration”) sells 500 cheese pizzas when the price is $10/pizza, and 1,100 cheese pizzas when the price is $8/pizza

a.  (3) Use these figures to calculate the price elasticity of demand for cheese pizzas.

b.  (3)  What is the marginal revenue associated with the additional 600 pizzas sold when the price is $8/pizza?

c.   (3)  Suppose the marginal cost of each pizza is $7/pizza.  What price should the pizzeria charge:  $8/pizza or $10/pizza?  Briefly explain why.
 

### a. Calculating the Price Elasticity of Demand (PED)
The price elasticity of demand (PED) measures the responsiveness of quantity demanded to a change in price. It is calculated using the formula:

\[
\text{PED} = \frac{\text{% Change in Quantity Demanded}}{\text{% Change in Price}}
\]

First, we need to find the percentage change in quantity demanded and the percentage change in price.

1. **Initial Price and Quantity:**
   - Initial Price, $ P_1 = 10 $
   - Initial Quantity, $ Q_1 = 500 $

2. **New Price and Quantity:**
   - New Price, $ P_2 = 8 $
   - New Quantity, $ Q_2 = 1100 $

3. **Calculate Percentage Changes:**
   - Percentage change in quantity demanded: 
     \[
     \frac{Q_2 - Q_1}{Q_1} = \frac{1100 - 500}{500} = \frac{600}{500} = 1.2 \text{ or } 120\%
     \]
   - Percentage change in price:
     \[
     \frac{P_2 - P_1}{P_1} = \frac{8 - 10}{10} = \frac{-2}{10} = -0.2 \text{ or } -20\%
     \]

4. **PED Calculation:**
   \[
   \text{PED} = \frac{1.2}{-0.2} = -6
   \]

Thus, the price elasticity of demand for cheese pizzas is $ -6 $, indicating a highly elastic demand.

### b. Marginal Revenue (MR)
Marginal revenue (MR) is the additional revenue that a firm earns from selling one more unit of a good. When we have discrete changes in quantity and price, we can use:

\[
\text{MR} = \frac{\Delta TR}{\Delta Q}
\]

where:
- $ \Delta TR $ = Change in Total Revenue
- $ \Delta Q $ = Change in Quantity

1. **Initial Revenue (at $10/pizza):**
   \[
   TR_1 = P_1 \times Q_1 = 10 \times 500 = 5000
   \]

2. **New Revenue (at $8/pizza):**
   \[
   TR_2 = P_2 \times Q_2 = 8 \times 1100 = 8800
   \]

3. **Change in Total Revenue:**
   \[
   \Delta TR = TR_2 - TR_1 = 8800 - 5000 = 3800
   \]

4. **Change in Quantity:**
   \[
   \Delta Q = Q_2 - Q_1 = 1100 - 500 = 600
   \]

5. **Marginal Revenue:**
   \[
   MR = \frac{3800}{600} = 6.33
   \]

The marginal revenue associated with the additional 600 pizzas sold at $8/pizza is $6.33 per pizza.

### c. Optimal Price Choice: $8/pizza or $10/pizza?
To decide whether the pizzeria should charge $8 or $10 per pizza, we compare marginal revenue (MR) with marginal cost (MC).

- **Marginal Cost (MC):** $7 per pizza.
- **Marginal Revenue (MR):** $6.33 per pizza when the price is $8/pizza.

Since $ MR < MC $ when the price is $8/pizza, selling additional pizzas at $8 results in a loss on each additional pizza sold.

**Conclusion:**  
The pizzeria should charge $10 per pizza. At this price, fewer pizzas are sold, but the revenue covers the marginal cost, maximizing profit.

### Follow-up Questions:
1. How would the PED change if the quantity sold increased more significantly with the price decrease?
2. What other factors should the pizzeria consider when setting prices, aside from marginal cost and revenue?
3. How would the marginal revenue change if the price dropped to $7/pizza?
4. What impact would a further decrease in price have on total revenue and profit?
5. If the pizzeria incurs additional fixed costs, how would this affect their optimal pricing decision?

**Tip:** Remember, when $ |PED| > 1 $ (elastic demand), a decrease in price leads to an increase in total revenue, while for $ |PED| < 1 $ (inelastic demand), a price decrease decreases total revenue.

Suppose a pizzeria, 'MPA = Master’s of Pizza Administration,' sells 500 cheese pizzas when the price is $10/pizza, and 1,100 cheese pizzas when the price is $8/pizza. a) Use these figures to calculate the price elasticity of demand for cheese pizzas. b) What is the marginal revenue associated with the additional 600 pizzas sold when the price is $8/pizza? c) Suppose the marginal cost of each pizza is $7/pizza. What price should the pizzeria charge: $8/pizza or $10/pizza? Briefly explain why.

Learn how to calculate the price elasticity of demand and marginal revenue for a pizzeria selling cheese pizzas at different price points. This problem demonstrates core concepts in economics such as elasticity, marginal revenue, and profit maximization, suitable for undergraduate economics students.

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